My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

The code contains #ifdefs to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL to produce the result for the original problem (default setting for most problems).

       
#include <iostream>
       
#include <vector>
       
 
       
#define ORIGINAL
       
 
       
// return (a*b) % modulo

       
unsigned long long mulmod(unsigned long long a, unsigned long long b, unsigned long long modulo)
       
{
       
#ifdef __GNUC__
       
  // use GCC's optimized 128 bit code
       
  return ((unsigned __int128)a * b) % modulo;
       
#endif
       
 
       
  // (a * b) % modulo = (a % modulo) * (b % modulo) % modulo
       
  a %= modulo;
       
  b %= modulo;
       
 
       
  // fast path
       
  if (a <= 0xFFFFFFF && b <= 0xFFFFFFF)
       
    return (a * b) % modulo;
       
 
       
  // we might encounter overflows (slow path)
       
  // the number of loops depends on b, therefore try to minimize b
       
  if (b > a)
       
    std::swap(a, b);
       
 
       
  // bitwise multiplication
       
  unsigned long long result = 0;
       
  while (a > 0 && b > 0)
       
  {
       
    // b is odd ? a*b = a + a*(b-1)
       
    if (b & 1)
       
    {
       
      result += a;
       
      result %= modulo;
       
      // skip b-- because the bit-shift at the end will remove the lowest bit anyway
       
    }
       
 
       
    // b is even ? a*b = (2*a)*(b/2)
       
    a <<= 1;
       
    a  %= modulo;
       
 
       
    // next bit
       
    b >>= 1;
       
  }
       
 
       
  return result;
       
}
       
 
       
// return (base^exponent) % modulo

       
unsigned long long powmod(unsigned long long base, unsigned long long exponent, unsigned long long modulo)
       
{
       
  unsigned long long result = 1;
       
  while (exponent > 0)
       
  {
       
    // fast exponentation:
       
    // odd exponent ? a^b = a*a^(b-1)
       
    if (exponent & 1)
       
      result = mulmod(result, base, modulo);
       
 
       
    // even exponent ? a^b = (a*a)^(b/2)
       
    base = mulmod(base, base, modulo);
       
    exponent >>= 1;
       
  }
       
  return result;
       
}
       
 
       
// Miller-Rabin-test

       
bool isPrime(unsigned long long p, bool fastCheckAgainstSmallPrimes = true)
       
{
       
  // IMPORTANT: requires mulmod(a, b, modulo) and powmod(base, exponent, modulo)
       
 
       
  // some code from             https://ronzii.wordpress.com/2012/03/04/miller-rabin-primality-test/
       
  // with optimizations from    http://ceur-ws.org/Vol-1326/020-Forisek.pdf
       
  // good bases can be found at http://miller-rabin.appspot.com/
       
 
       
  // trivial cases
       
  if (fastCheckAgainstSmallPrimes)
       
  {
       
    const unsigned int bitmaskPrimes2to31 = (1 <<  2) | (1 <<  3) | (1 <<  5) | (1 <<  7) |
       
                                            (1 << 11) | (1 << 13) | (1 << 17) | (1 << 19) |
       
                                            (1 << 23) | (1 << 29); // = 0x208A28Ac
       
    if (p < 31)
       
      return (bitmaskPrimes2to31 & (1 << p)) != 0;
       
 
       
    if (p %  2 == 0 || p %  3 == 0 || p %  5 == 0 || p % 7 == 0 || // divisible by a small prime
       
        p % 11 == 0 || p % 13 == 0 || p % 17 == 0)
       
      return false;
       
 
       
    if (p < 17*19) // we filtered all composite numbers < 17*19, all others below 17*19 must be prime
       
      return true;
       
  }
       
 
       
  // test p against those numbers ("witnesses")
       
  // good bases can be found at http://miller-rabin.appspot.com/
       
  const unsigned int STOP = 0;
       
  const unsigned int TestAgainst1[] = { 377687, STOP };
       
  const unsigned int TestAgainst2[] = { 31, 73, STOP };
       
  const unsigned int TestAgainst3[] = { 2, 7, 61, STOP };
       
  // first three sequences are good up to 2^32
       
  const unsigned int TestAgainst4[] = { 2, 13, 23, 1662803, STOP };
       
  const unsigned int TestAgainst7[] = { 2, 325, 9375, 28178, 450775, 9780504, 1795265022, STOP };
       
 
       
  // good up to 2^64
       
  const unsigned int* testAgainst = TestAgainst7;
       
  // use less tests if feasible
       
  if (p < 5329)
       
    testAgainst = TestAgainst1;
       
  else if (p < 9080191)
       
    testAgainst = TestAgainst2;
       
  else if (p < 4759123141ULL)
       
    testAgainst = TestAgainst3;
       
  else if (p < 1122004669633ULL)
       
    testAgainst = TestAgainst4;
       
 
       
  // find p - 1 = d * 2^j
       
  auto d = p - 1;
       
  d >>= 1;
       
  unsigned int shift = 0;
       
  while ((d & 1) == 0)
       
  {
       
    shift++;
       
    d >>= 1;
       
  }
       
 
       
  // test p against all bases
       
  do
       
  {
       
    auto x = powmod(*testAgainst++, d, p);
       
    // is test^d % p == 1 or -1 ?
       
    if (x == 1 || x == p - 1)
       
      continue;
       
 
       
    // now either prime or a strong pseudo-prime
       
    // check test^(d*2^r) for 0 <= r < shift
       
    bool maybePrime = false;
       
    for (unsigned int r = 0; r < shift; r++)
       
    {
       
      // x = x^2 % p
       
      // (initial x was test^d)
       
      x = mulmod(x, x, p);
       
      // x % p == 1 => not prime
       
      if (x == 1)
       
        return false;
       
 
       
      // x % p == -1 => prime or an even stronger pseudo-prime
       
      if (x == p - 1)
       
      {
       
        // next iteration
       
        maybePrime = true;
       
        break;
       
      }
       
    }
       
 
       
    // not prime
       
    if (!maybePrime)
       
      return false;
       
  } while (*testAgainst != STOP);
       
 
       
  // prime
       
  return true;
       
}
       
 
       
int main()
       
{
       
  // generate all primes up to 500
       
  std::vector<unsigned int> primes = { 2 };
       
  for (unsigned int i = 3; i < 500; i += 2)
       
  {
       
    bool isPrime = true;
       
 
       
    // test against all prime numbers we have so far (in ascending order)
       
    for (auto x : primes)
       
    {
       
      // prime is too large to be a divisor
       
      if (x*x > i)
       
        break;
       
 
       
      // divisible => not prime
       
      if (i % x == 0)
       
      {
       
        isPrime = false;
       
        break;
       
      }
       
    }
       
 
       
    // yes, we have a prime
       
    if (isPrime)
       
      primes.push_back(i);
       
  }
       
 
       
  unsigned int tests = 1;
       
#ifndef ORIGINAL
       
  std::cin >> tests;
       
#endif
       
  while (tests--)
       
  {
       
    // largest number to be considered
       
    unsigned int limit = 150*1000000;
       
    std::cin >> limit;
       
 
       
    // always six consective increments where n^2 + good[i] must be prime
       
    std::vector<unsigned int> good = { 1, 3, 7, 9, 13, 27 }; // will be overwritten by Hackerrank input
       
#ifndef ORIGINAL
       
    for (auto& x : good)
       
      std::cin >> x;
       
#endif
       
 
       
    bool sameParity = true;
       
    unsigned int parity = good.front() % 2;
       
    for (auto x : good)
       
      sameParity &= x % 2 == parity;
       
    if (!sameParity)
       
    {
       
      std::cout << "0" << std::endl;
       
      continue;
       
    }
       
 
       
    // collect all numbers between 1 and the largest element of good which are not part of good
       
    // => n^2 + bad[i] must not be prime
       
    std::vector<unsigned int> bad;
       
    for (unsigned int i = parity; i < good.back(); i += 2)
       
    {
       
      bool isGood = false;
       
      for (auto x : good)
       
        isGood |= x == i;
       
 
       
      if (!isGood)
       
        bad.push_back(i);
       
    }
       
 
       
    // performance tweak: if good[] contains 1, 3, 7 and 9 then n must be a multiple of 10
       
    // or in other words: bad[0] = 5, bad[1] > 9
       
    unsigned int increment = 2;
       
    unsigned int start     = 1 - parity;
       
    if (bad.size() >= 2 && bad[0] == 5 && bad[1] > 9)
       
      increment = 10;
       
 
       
    // sum of all matching numbers
       
    unsigned long long sum = 0;
       
    for (unsigned int n = start; n < limit; n += increment)
       
    {
       
      auto square = n * (unsigned long long)n;
       
 
       
#ifdef ORIGINAL
       
      // n^2 + 3, n^2 + 7 and n^2 + 13 must be primes
       
      // => n^2 must not be divisible by 3, 7 or 13
       
      if (square % 3 == 0 || square % 7 == 0 || square % 13 == 0)
       
        continue;
       
#endif
       
 
       
      // check all numbers against small primes
       
      bool ok = true;
       
      for (auto p : primes)
       
      {
       
        for (auto check : good)
       
        {
       
          auto current = square + check;
       
          // not prime if remainder is zero
       
          if (current != p && current % p == 0)
       
          {
       
            // note: for n=10, current can be part of primes[],
       
            // that's why I check "current != p" to prevent false negatives
       
            ok = false;
       
            break;
       
          }
       
        }
       
 
       
        // failed (= not prime) ?
       
        if (!ok)
       
          break;
       
      }
       
 
       
      // use slower Miller-Rabin if still ok
       
      for (auto x : good)
       
        if (ok)
       
          ok &=  isPrime(square + x, false);
       
      for (auto x : bad)
       
        if (ok)
       
          ok &= !isPrime(square + x, true);
       
 
       
      // passed all tests ?
       
      if (ok)
       
        sum += n;
       
    }
       
 
       
    // display result
       
    std::cout << sum << std::endl;
       
  }
       
 
       
  return 0;
       
}

This solution contains 42 empty lines, 60 comments and 11 preprocessor commands.